1. Distributed Models of Drug Kinetics
Paul F. Morrison Ph.D.
Drug Delivery and Kinetics Resource,
Division of Bioengineering and Physical Science, NIH

2. Previous: Whole organ pharmacokinetic models
® C(t), Cp(t)
Multiple organs
Multiple chemical species
Today: Distributed models of pharmacokinetics
® C(x, t)

3. Outline General principles behind distributed models
Simple formalism
Major physiological-physical factors
Examples where distributed kinetics are important in drug delivery
Peritoneal cavity
Intraventricular infusion
Direct interstitial infusion in the CNS
Microdialysis
Delivery of tight-binding antibodies

4. General Principles
Central issue: What is the situation which leads to a spatially-dependent distribution of drug in a tissue, and how is this distribution described quantitatively?
Essential Characteristic: Target tissue can not be approximated as a well-mixed compartment. Specifically, drug is delivered along a path from the source along which local clearance or binding causes the concentration to vary.

5. Common Situations Leading to Spatial Distributions
Delivery of agents from spatially non-uniform sources
e.g. point source of needle tips
e.g. surface (planar) sources encountered when drug solution bathes the surface of a target organ
e.g. intravenous delivery into tissues with highly heterogeneous capillary densities
Certain tumors
Delivery of very highly binding substances leading to a “moving front” concentration boundary
High affinity antibody conjugates

6. Distributed Capillary Bed

7. Diffusion-Reaction Formalism
Mass balance for substance in tissue: 2 C C k m æ C ö = D - C - pa ç - C ÷ p 2 t è R ø x R
rate of
conc
net
diffusion
metabolism
net
transport
change in D V in D V in D V
across
microvasculature If C ( t ) , c c p - D DV - D x x C ( x , ) = , IC x x + D x , C ( ) t = R C
inf , BC
kmC /R C ( , t ) = , BC
pa(C/R-Cp)

8. (Diffusion-reaction Cont’d)
Then C ( x , t ) 1 é - x ù é x ù =
exp
erfc - kt ê ú ê ú R C 2 ë k / D û ë 4 Dt û
inf 1 é x ù é x ù +
exp
erfc + kt ê ú ê ú 2 ë k / D û ë 4 Dt û ( ) k m + pa
where k = / R
At steady state, this simplifies to just: C ( x ) [ ] = R
exp - x k / D C
inf
If no reaction is present, then: C ( x , t ) é x ù = R
erfc ê ú C ë 4 Dt û
inf

9. Implications
Drugs can be delivered to tissue layer near exposed surface but the thickness of this layer depends strongly on metabolism of agent
Delivery of non-metabolized agents across surfaces for purposes of systemic drug administration (e.g. I.P.) is dominated by distributed microvascular uptake in the tissue layer underlying the surface
And not by the rate of transfer across the peritoneal membrane per se

10. AUC = ò C ( t ) dt is NOT applicable , instead AUC ( x ) = ò C ( x , t
Importantly, local dose-response relationships also become spatially dependent, e.g.
AUC = ò C ( t ) dt is
NOT
applicable ,
instead
AUC ( x ) = ò C ( x , t ) dt is
applicable .

11. Diffusion Constants and Reaction Rate Constants
Diffusion constants extrapolated from known values for reference substances
Reaction rate constants
Cultured cell metabolic data
Fittings of compartment models to whole tissue data
From autoradiography and fit to exp[-x√(k/D)] - 2 37 2 D = l * D l a * . 6 a ( MW )
tissue
aqueous æ ö . 6 D æ MW ö
tissue
ref ç ÷ = ç ÷ ç ÷ D è MW ø è ø
tissue ,
ref

12. (continued) pa µ MW so æ ö pa æ MW ö ç ÷ = ç ÷ ç ÷ pa è MW ø è ø -
Concentration profile insensitive to MW if k= pa/R (i.e. inulin depth ~ urea) - 63 . pa MW so æ ö . 63 pa æ MW ö
ref ç ÷ = ç ÷ ç ÷ pa è MW ø è ø
ref

13. Example 1 Intraperitoneal Administration of Agents
Treatment of ovarian cancer patients

14. The ovarian goals:
Achieve sufficient penetration of surfaces to treat resident tumor nodules
Nodules lie on serosal surface, not invasive
Not metastatic
Diameter of <5 mm (in 73% of post surgical cases)
Complete irrigation of serosal surfaces
Predicted pharmacokinetic advantage of I.P. over I.V. delivery
Alberts et al confirmed survival advantage for patients treated
with cisplatin

15. Predicted Penetration in Peritoneum
Cisplatin
Cis-diamminedichloroplatinum (II)
Slow hydrolysis
Compute C(x) at steady state
How good is the model?
Estimate predictability of diffusion model for surrogate EDTA

17. Concentration Profile of EDTA at Peritoneal Interface of GI Tissue
0.01
0.1 1 C - f e p ( )
inf =
exp x k / D [ ]
where . 26 ,
0078 m 1
Relative Tissue Concentration
C / Cinf
Plasma
200
400
600
800
1000
Distance from Peritoneum (µ)

18. Example 2 Intraventricular Delivery
-- Attempt to circumvent the BBB

19. Blood-Brain Barrier Exclusion of Histamine
Pardridge et al, Ann Int Med 105; (1986)

20. Sucrose Distribution After Intraventricular Administration
Groothuis et al, J. Neurosurg. 1998

21. Similar Penetration with Macromolecules
For macromolecules (MW > 67kD), D = fe De /R
decreases by >10-fold and effective pa decreases by a similar amount. Thus, in the capillary permeation limit ln 2
Penetration
depth ( d ) = p pa + k m

22. Intraventricular Infusion of BDNF
Yan et al, Exp. Neurol. 127; (1994)

23. Delivery of NGF from Implanted Polymer
2.5mm
2.5mm
Krewson et al, Brain Res. 680; (1995)

24. Example 3 Direct Interstitial Infusion (microinfusion)
-- approach to achieve more widespread distribution

25. Direct Infusion Cannula in Brain

26. On a cellular scale:
Slow flow:
0.5 to 10 µl/hour
(Alzet range)
High flow:
0.1 to 4.0 µl/min
(Harvard pump)

27. Two Techniques for Direct Interstitial Infusion
Alzet pump
Harvard Pump
250 µl
10 µl
Syringe system

28. (0.9 µl/hr of cisplatin for 160 hrs)
Slow Microinfusion
(0.9 µl/hr of cisplatin for 160 hrs)
Applicable mass balance:
Steady state solution for continuous infusion,
after matching mass inflow rate to
diffusive flux at cannula tip, is:
qCinf is mass infusion rate, and D=f De/R.
R= 1 for cisplatin, R=f for IgG, where f is
the extracellular volume fraction. Time to
steady state at r = 4 mm is 3 hrs for cisplatin.

29. Concentration Profile of Cisplatin in Rat Brain (0.9 µl/hr)

30. Issues Treatment volumes r >1 cm
If pumping rates taken to maximum range, how does the response of high-flow delivery compare with low-flow (diffusional) delivery?
Simple estimators of profiles and volumes of distribution

31. High Flow Infusion Modelp
Pressure:
Darcy’s Law:
Continuity Equation for Water
Minor deformation
Macromolecular mass balance: i v ( r ) = - k r C 1 C 1 C k m 2 2 = D r - r v C - pa ( - C ) - C 2 2 p t r r r R r r R R
change
net
diffusion
net
bulk
flow
net
mass
transfer
metabolic in
total
into a
volume
into a
volume
across
capillaries
loss
conc in D t
element
element

32. Simplification of Mass Balance
Concentration Profile:
where q Cinf is the mass infusion rate, ro is the cannula radius
Penetration depth at S.S. and time to reach it:
1.8 cm days hr=> km
3.6 cm days infinity => km

34. Concentration Profiles in Brain (3 µl/min)

35. Brain Infusion Technology
Delivery
Predicted distribution of albumin-aCSF in
striatum from flow-diffusion-reaction equation

36. 111-In-DTPA-Transferrin Infusion

37. MODEL: POST-INFUSION PHASE
Pure Diffusional Relaxation of
End-of-Infusion Profile
Profile: C ( r , t ˆ ) = - k t ˆ e 2 ˆ [ - ( r - r ) /( 4 D t ) - ( r 2 + r ) /( 4 D t ˆ ) ò C ( r , t ) e - e ] r d r 2 r D t ˆ
inf p t ˆ
>
where D = f D / R , k = ( pa + k ) / R , t ˆ = t - t e m
inf

38. Post-infusional Behavior of Concentration Profiles

39. Post-infusional Behavior of Concentration Profiles

40. Increased Treatment Volume with High Flow Infusion

42. Application: Chemical Pallidotomy
Parkinson’s disease
Alternative to thermal ablation
Problem of optic nerve destruction
Quinolinic acid
Parameters from microdialysis
Small molecular weight

43. Simulated Quinolinic Acid Profiles in Gpi Following Direct Interstitial Infusion

44. Chemopallidectomy

45. Chemopallidectomy
Gpi, Intact Side
Gpi, Lesioned Side

46. Summary General principles of distributed pharmacokinetics Examples
-- Peritoneal
-- Intraventricular
-- Interstitial
Next week: Population pharmacokinetics